Aims: We present a multiwavelength study of the Be/X-ray binary system SAX J2103.5+4545 with the goal of better characterizing the transient behaviour of this source. Methods: SAX J2103.5+4545 was observed by Swift/XRT four times in 2007 from April 25 to May 5, and during quiescence in 2012 August 31. In addition, this source has been monitored from the ground-based astronomical observatories of El Teide (Tenerife, Spain), Roque de los Muchachos (La Palma, Spain), and Sierra Nevada (Granada, Spain) since 2011 August, and from the TÜBİTAK National Observatory (Antalya, Turkey) since 2009 June. We performed spectral and photometric temporal analyses to investigate the different states exhibited by SAX J2103.5+4545. Results: In X-rays, an absorbed power-law model provided the best fit for all the XRT spectra. An iron-line feature at ~6.42 keV was present in all the observations except for that taken during quiescence in 2012. The photon indexes are consistent with previous studies of SAX J2103.5+4545 in high/low-luminosity states. Pulsations were found in all the XRT data from 2007 (2.839(2) mHz; MJD 54 222.02), but not during quiescence. The two optical outbursts in 2010 and 2012 lasted for about eight or nine months (as the one in 2007 probably did and the current one in 2014 might do) and were most probably caused by mass-ejection events from the Be star that eventually fed the circumstellar disc. All of these outbursts started about three months before the triggering of the X-ray activity, and at about the same period before the maximum of the Hα line equivalent width (in emission) was reached at only ~-5 Å. The global correlation between the BV variability and the X-ray intensity was also observed at longer wavelengths in the IR domain. Tables 5 and 6 are available in electronic form at http://www.aanda.org